Introduction to realizability of modules over Tate cohomology
David Benson, Henning Krause and Stefan Schwede

\begin{abstract}
This paper is a companion to our paper, ``Realizability of modules over
Tate cohomology,'' in which we described an obstruction theory which
applies in a number of contexts. This companion paper restricts attention to
Tate cohomology, and gives constructions and proofs within that framework.
A special case of our main theorem is as follows. 
Let $k$ be a field and let $G$ be a finite group. There is a canonical
element in the Hochschild cohomology of the Tate cohomology
$\gamma\in\HH^{3,-1}\hat H^*(G,k)$ 
with the following property. Given
a graded $\hat H^*(G,k)$-module $X$, the image of $\gamma$
in $\Ext^{3,-1}_{\hat H^*(G,k)}(X,X)$ vanishes if and only if $X$ is 
isomorphic to a direct summand of $\hat H^*(G,M)$ for some $kG$-module $M$. 
If $X$ is realizable in this way, then the essentially different ways
of realizing it form an affine space whose associated vector space is
$\Ext^{2,-1}_{\hat H^*(G,k)}(X,X)$.
\end{abstract}